In a coordinated cellular system, a DA arrangement may employ various schemes to control other-cell interference. In one approach, Costa-precoding (also known as dirty-paper coding (DPC)) followed by zero-forcing beamforming (ZFBF) may be employed to regulate downlink signals transmitted to users so as to minimize mutual interference among users of respective cell origins. In this scheme, a network produces the signals to be transmitted to different users in a pre-determined order (e.g., a signal for user 1 is produced first, a signal for user 2 is produced next, etc.). A significant constraint with a DPC-ZFBF scheme is that a signal transmitted to, for example, user (i), must not create interference at the antennas of all other users who preceded user (i) according to the pre-determined order.
A cellular environment may include, among other things, a network having (t) transmitters, and (m) mobile devices having (r) receiving antennas per mobile device. For example, the cellular environment may include users (i) (i=1, 2, . . . , m) corresponding to the mobile devices. Each mobile device may be in a different cell, and the transmitters can send independent messages to the mobile devices. One or more of the transmitters may not be co-located.
For purposes of discussion, assume there is an average total power constraint (P) at the transmitters. The Gaussian broadcast channel (GBC) may be an additive noise channel and each time sample can be represented by the following expression:yi=Hix+ni i=1, 2, . . . , m,  (1)where (x) is a vector of size (t*1) that represents the total signal transmitted from all of the transmitters. Under a total average power constraint at the transmitters, it may be required that the E[x†x]<P. Yi is the output vector received by users (i). The output vector is a vector of size (r*1). Hi is a fixed matrix channel for users (i) whose size is (r*t). These channel matrices are fixed and known at the transmitters and the mobile devices. Ni is a Gaussian, circularly symmetric, complex-valued random noise vector with a zero mean and a covariance of σ2iI.
For purposes of discussion, assume that the total number of transmit antennas equals the total number of receive antennas (i.e., t=r*m), and that (r) independent streams are transmitted to each mobile device. Thus, t=r*m independent streams may be transmitted in total. Additionally, let (xj) denote the symbols of the j-th transmitted stream with power (qj), and (Φi)={j|xj belongs to users (i)}. Associated with each transmitted stream (xj) is a transmitted beamforming vector (Vj). Thus, the total transmitted signal may be represented by the following expression:
                    x        =                              ∑                          j              =              1                        i                    ⁢                                          ⁢                                    x              j                        ⁢                                          V                j                            .                                                          (        2        )            
Assume that (Vj) has unit norm. The signal transmitted for users (i) may be represented by the following expression:
                                          x            i                    =                                    ∑                              j                ∈                                  Φ                  i                                                      ⁢                                                  ⁢                                          x                j                            ⁢                              V                j                                                    ,                            (        3        )            and the covariance of the signal transmitted for users (i) may be represented by the following expression:
                                          S            i                    =                                    ∑                              j                ∈                                  Φ                  i                                                      ⁢                                          V                i                            ⁢                              V                j                †                            ⁢                              q                j                                                    ,                            (        4        )            and the covariance of the total transmitted signal x may be represented by the following expression:
                    S        =                              ∑            i                    ⁢                                          ⁢                                    S              i                        .                                              (        5        )            